In this work we will generalize the results developed in ref.[1]} for motion in hypersurface \(V_{(N-1)}\)  to \(V_L ({1\le L < N})\)  in N dimensional Euclidean space \(R^N\) . As we know, the quantum mechanical analysis of the system in curved space has been examined about the ordering problem for a long time [2]}, [3]}. Also, the problem of the quantization of a dynamical system constrained to a curved manifold embedded in the higher-dimensional Euclidean space has been extensively investigated as one of the quantum theories on a curved space [4]}, [5]}, [6]}. For the ordering problem primarily, two approaches have been used, canonical quantization and path-integral method. Here we are taking a non-relativistic particle constrained to a curved surface embedded in the higher dimensional Euclidean space [7]}, [8]}. These type of systems and their various properties such as quantization in different approaches, their comparisons and applications have been studied by many authors [9]}, [10]}. This has motivated us to construct the BRST symmetry for this kind of system. Here for the first time we have tried to BRST quantize the system of a non-relativistic particle constrained to a curved surface embedded in the higher dimensional Euclidean space in both Hamiltonian and Lagrangian formalism.
